Lozanovsky’s Notebooks (Part II)

301 Pages·2012·16.503 MB·English

by  Lozanovsky, G.Y. (ed Wójtowicz, M.)

l*füt
Problems 610-1469
(Notebooks IV—XH)
AFILIACJE I KLASYFIKACJA NAUKOWA
Marekwojtowicz
Institute ofMathematics
Casimirthe GreatUniversity
85-072 Bydgoszcz
POLAND
[email protected]
Lech Maligranda Mieczyslaw Mastylo
DepartmentofEngineering Sciences Faculty ofMathematics
and Mathematics and Computer Science
LuleaUniversity ofTechnology A. MickiewiczUniversity
SE—971 87 Lulez‘i 61-614 Poznan
SWEDEN POLAND
[email protected] [email protected]
WitoldWnuk DavidYost
Faculty ofMathematics Centre forInformatics
and Computer Science andApplied Optimization
A. Mickiewicz University University ofBallarat
ul. Umultowska 87 PO Box 663
61-614 Poznan Ballarat, Vic. 3353
POLAND AUSTRALIA
[email protected] [email protected]
Mathematics SubjectClassification (2010): 00A07,46Bxx,46Exx, 47B65
Problems 610—1469
(Notebooks IV—-XH)
Translated with collaboration of
from the Russian Dorota Wéjtowicz
and scientific: comments by
and edited by
Lech Maligranda
Marek Wéjtowicz
Mieczyslaw Mastylo
Witold Wnuk
David Yost
KomitetRedakcyjny
JanuszOstoja-Zagélski (przewodniczqcy)
Katarzyna Domafiska, RyszardGerlach, SlawomirKaczmarek
PiotrMalinowski,JacekWoiny, GraZynaJarzyna (sekretarz)
ProjektOkladkz‘
P.M. LOGO
© CopyrightbyWydawnictwo UniwersytetuKazimierzaWielkiego
Bydgoszcz2012
© Rita Lozanovskaya
© L. Maligranda, M. Mastyio, W. Wnuk, M. Wéjtowicz,D. Yost
UtwérniemoZebyépowielanyirozpowszechnianywca1oécianiwefragmentach
bezpisemnej zgodyposiadaczaprawautorskich
ISBN 978-83-7096—855—7
Wydawnictwo UniwersytetuKazimierzaWielkiego
(Cz1onekPolskiej IzbyKsiqiki)
85-092Bydgoszcz,ul. Ogifiskiego 16
tel/fax[phone],mail:[email protected]
http://www.wydawnictwo.ukw.edu.pl
Drukioprawa:DrukamiaTOTEM
88-1001nowroc1aw,ul. Jacewska89
tel.[phone]
mail: [email protected]
Poz. 1407.Ark. wyd. 16
Contents
Notations and terminology ........................................ 6
Preface ............................................................. 11
Notebook IV (problems 610—715a) .............................. 13
Notebook V (problems 715—866) .................................45
Notebook VI (problems 867—981) ................................85
Notebook VII (problems 892—1075) ............................ 116
Notebook VIII (problems 1076—1144) ..........................146
Notebook IX (problems 1145—1235) ............................ 174
Notebook X (problems 1236—1317) .............................201
Notebook XI (problems 1318—1395) ............................231
Notebook XII (problems 1396—1468) ...........................256
List of publications of Grigorii Ya. Lozanovsky ............... 279
Other papers quoted in this book .............................. 285
Notations and terminology
(...?) lllegible handwriting
(...)
Inessential text
[?l Unknown reference or argument
(LM),
(MM), (WW), (MW), (DY) — comment(s) to a problem made by Lech Ma-
ligranda, MieczyslawMastylo, WitoldVVnuk, MarekW6jtowicz, DavidYost,
respectively
£(X,Y)
The space of continuous operators from a normed space X into a normed
space Y
Xik The space £(X,R)
£(X) The space £(X,X)
5,,(X,Y)
The space of order bounded operators from a linear lattice X into a linear
lattice Y
we Y)
The space of regular operators from a linear lattice X into a linear lattice
Y (if Y is Dedekind complete, then £r(X,Y) = £b(X,Y))
£r(X)
The space £T(X,X)
£C(X,Y) The space of cr—order continuous operators from a linear lattice X into
a linear lattice Y
£n(X,Y) The space oforder continuous operators from a linear lattice X into a linear
lattice Y
£n(X) The space £n(X,X)
5(T,Elli) The space of all [equivalence classes of] measurable functions on a measure
space (T72):u)
s, 3[o,1]
The space of all [equivalence classes of] Lebesgue measurable functions
on [0, 1]
X
The lattice £r(X,R); if X is a Banach lattice then X = X*
6
l The order continuous dual of X — see Vulikh [178, Def._IX.2.1] - the space
<
>
Ln(X,11)); Lozanovsky used only the Vulikh’s symbol X for fin(X,R), and
not the classical symbol X,I
XI The Kothe dual of a Banach function space X [i.e., an order dense ideal
of S(T,)3,u), and the norm ofX is monotone with respect to the ordering
a.e.]: X’ = {y E S(T,ZI,,u) : cry 6 L1(T,E,,u) for all :1: e X}
The anormal [2 singular] part ofX
jam 523
The space {1 = $1 -$2 : $1 6 E1,:52 e E2}, where E,- is a linear subspace
E10 E2
of S[0,1], i = 1,2
Universal completion of an Archimedean lattice X
Distance between two closed subspaces P and Q of a given Banach space
X, i.e., the Hausdorffdistance between the unit balls Bp and BQ of P and
Q, respectively:
9(P,Q) 2: max{sup{d(z,Bq) : a: e Bp},sup{d(y,Bp) 2 y 6 Egg”
mnT+00 The sequence (In) in a linear lattice X is not order bounded
PA The order projection (if it exists) from a linear lattice X onto the band
generated by a nonempty subset A of X; if A : {1:}, then P1 denotes PM}
(X**)1r The set of all the elements 1:“ ofthe bidual X** ofa Banach space X such
that, there is a family {xtheT in X satisfying the following two conditions:
(a) ZtET lF(7r($t))l < 001 VF E X***)
(b) Zt E T2:*(:ct) = x**(m*), V:1:* E X*.
Here1r denotesthecanonicalembeddingofX intoX**. See [L18, Definitions
1 and 3].
(x**)1r The set of all the elements :13“ ofthe bidual X** of a Banach space X such
that, ZteT:c**(:v,’§) = 0 for every family {mi} in X* satisfying the following
two conditions:
(a) ZtET Iy**(x;‘)l < 00, W“ e X“,
(b) ZtET33:(13) = 0, V1: 6 X. See [L18, Definitions 2 and 3].
The Banach lattice ofthe form X23 2 {:12 6 S'[O, 1] : |$|lD E X}, endowedwith
the norm “mHX’ := H |x|P“1/P, where X = (X, H H) is a Banach lattice and,
at the same time, an order ideal of S[0, 1]
(or Xu) The principal ideal of a linear lattice X generated by u E X:
X(u) :2 {3; e X: III 3 M11] for some A > 0};
it is an M-space with the norm ||:z:||(u) :2 inf{)\ > 0 : |:1:| S A|u|}
7
(*)(a)—convergence — if in an object X there is defined an (a)—convergence for
sequences of X: {n g £. then the (*)(a)-convergence is defined as follows:
fin (fl) é ifffor every subsequence (fink) there exists a subsequence (6“,) with
(a)
{BL-I __' g
(ru)-convergence — Convergence with a regulator: In L)» a: in a linear lattice
X iff I mn — :3 lg enr for some sequence ofreals en ——> O and r E X+
(bo)-linear operator [178, Ch. VIII, §6] — an operator A acting between two
linear lattices X and Y, with X a normed lattice, such that the condition
13,, u 3: implies A2,, 12> Am
Abstract norm ofan operator — ifX is aBanach lattice, Y is a K+-space, and
A : X —> Y is a. (bo)-linearoperator, then the value “All = sup{|A$[ : “at” g 1}
exists in Y and is called the abstract norm of A
Almost integral operator — every element of £r(X,Y), where X,Y are KB—
spacos belonging tp the band generated by the class of all finite-dimensional
range elements of LAX,Y)
The basis of a linear lattice X with unit element 6 — the Boolean algebra [178,
Def. III.12.2]
B(e)={$EX::v/\(e—z)=0}
Countability of type of a linear lattice X (the countable sup property) —
every order bounded and pairwise disjoint family of positive elements of X is
at most countable
Fréchet lattice — a metrizable locally convex-solid linear lattice (X,7') that is
in addition T-complete; the topology 7' is then generated by a countable family
{9n}:o of Riesz seminorms or, equivalently, by the Riesz metric
“ ’ ”=12" 1+ Qn($»y)
K+-space ([178, Ch. VI, §6] cf. [140, pp. 51-52]) — a Dedekind complete linear
lattice X such that a nonempty subset D C X is order bounded iff for every
sequence (In) C D and every sequence (An) of real numbers with A" —> 0
we have that Anzn fl» 0; the lattices S[0,1], Zoo, and all Orlicz lattices are
K+-spaces ([140], pp. 51-52; [132], Theorem 3.2)
KB‘-space ([178]) — a FYéchet lattice (X,T) in which every increasing sequence
(:rk) inX+ isT-convergentprovidedthatitistopologicallyboundedinX; equiv-
alently, theRiesz metric gonXdefined byasequence (9,.) ofRieszseminormsis
both monotonecomplete (i.e., 3;; T +00 with k —> 00 iiilin1k_.°°gn(:ck) = 00 for
some n E N) and a—order continuous; every KB*-space is Dedekind complete
Locally solid topology — a. linear topology ’7‘ on a linear lattice X with a basis
of zero consisting of solid sets; if T is at the same time locally convex, then it
is called locally-convex solid (every such topology is generated by a family
{90} of Riesz (:monotone) seminorms; see [6, Theorem 6.1])
The maximal normed extensionofa normed sublattice X ofS(,u) —the lattice
[X] == {1‘ E SW) 1 ”II [X] == supfllgllx =9 e [0. [fl] flX} < oo},

endowed with the norm H “[X].
Minihedral cone—aconeK inavectorspaceX suchthat, foreverypair1:,y E X
there exists 2 E X such that z+K = (:r+ K) 0(y+K); ifX is a linear lattice,
then the cone K = X+ is minihedral since (a: + K) H (y + K) = a: V'y + K
MS-sequence — Moore-Smith sequence: a function x z A —> X from a directed
set A (without the greatest element) into X, denoted often as {320, : a E A}
Nakano reflexive linear lattice — a linear lattice X satisfying the identity (X) =
X, i.e., (X3); = X
Nakano subspace ofa Dedekind complete linear lattice X — asublattice Y ofX
such that, infT e Y for every nonempty subset T of Y+, where the infimum is
taken in X (i.e., Y is a regular sublattice ofX [6, p. 6])
quasi-(r)-complete linear lattice [L29, Def. 4] — an Archimedean linear lattice
X with the following property: for every finite sequence g = {mi}?=1 G X,
n = 1,2,..., and u := |:1:1| 4- |$2| + + |:r,,|, the II "(w-closure X(£) of the
sublattice of X01) generated by g is I] ”(w-complete; every (r)-complete linear
lattice is quasi-(r)-complete, but not conversely [L29, Examples 1,2]
Stable-convergent sequence [178, Ch. VI, §4] — a sequence (13") in a linear
lattice X such that :
(1) (mn) is (o)—convergent to O, and
(2) there exists a sequence (An) ofreals with limAn = 00 and Anxn (:2 O
A sequence (:cn) C L1[0, 1] with uniformly absolutely continuous integrals:
for every 6 > 0 there exists 6 > 0 such that, for every n E N and E e E the
condition /\E < 6 implies that f53,;(t)d/\(t) < 6; here 2 denotesthealgebra of
all [classesof] Lebesgue measurablesubsetsof[0,1], and A denotesthe Lebesgue
measure 011 [0,1]
Universally complete linear lattice — a linear lattice that is both laterally and
Dedekind complete